Characterizations of matrix equalities involving the sums and products of multiple matrices and their generalized inverse

It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $A_1{B}_1^{g_1}{C}_1+A_2{B}_2^{g_2}{C}_2+\cdots +A_k{B}_k^{g_k}{C}_k=D$ and $A_1{B}_1^{g_1}{A}_2{B}_2^{g_2}\cdots A_k{B}_k^{g_k}{A}_{k+1}=A$, where $A_1$, $A_2$, $\dots$, $A_{k+1}$, $C_1$, $C_2$, $\dots$, $C_k$ and $A$ and $D$ are given, and $B_1^{g_1}$, $B_2^{g_2}$, $\dots$, $B_k^{g_k}$ are generalized inverses of matrices $B_1$, $B_2$, $\dots$, $B_k$. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $\{1\}$- and $\{1,2\}$-generalized inverses of matrices with $k=2,3$ by using some elementary methods, including a series of explicit rank equalities for block matrices.